Four-Bar Linkage Mechanism Vector Loop Equations

Four-Bar Linkage Vector Loop Equations

The derivation of the vector loop equation for a four-bar linkage mechanism involves defining the position vectors of the links and the angles between them. The four-bar linkage consists of four links connected by four joints, forming a closed loop as shown in the figure below.

Vector Loop Equation

The vector loop equation is derived by summing the position vectors around the closed loop of the four-bar linkage and setting their sum to zero:

\[\mathbf{R}_A+\mathbf{R}_{BA}-\mathbf{R}_{BO_4}-\mathbf{R}_{O_4}=0\]

To obtain an alternative form of the equations, often used in analysis, complex number notation can be used for the position vectors. Using complex notation, the following is obtained

\[a e^{j\theta_2}+b e^{j\theta_3}-c e^{j\theta_4}-d e^{j\theta_1}=0\]

where $a$, $b$, $c$ and $d$ represent the scalar length of each of the links.

Euler’s equation, $e^{j\theta} = (\cos \theta + j \sin \theta)$ may also be substituted into the equations to obtain this form

\[a(\cos\theta_2+j\sin\theta_2)+b(\cos\theta_3+j\sin\theta_3)-c(\cos\theta_4+j\sin\theta_4)-d(\cos\theta_1+j\sin\theta_1)=0\]